Skip to main content
SpringerLink
Log in

On Radial Functions and Distributions and Their Fourier Transforms

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We give formulas relating the Fourier transform of a radial function in \(\mathbb{R}^{n}\) and the Fourier transform of the same function in \(\mathbb{R}^{n+1}\), completing the analysis of Grafakos and Teschl (J. Fourier Anal. Appl. 19:167–179, 2013) where the case of \(\mathbb{R}^{n}\) and \(\mathbb{R}^{n+2}\) was considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In [9] a=−2π, while in this article we take a=1. The choice a=−1 is also popular. Once the formulas are known for a particular value of a the formulas for other choices are easy to derive.

  2. One can also work in the spaces \(\mathcal{D} ( \mathbb{R}^{n} ) \) and \(\mathcal{D}^{\prime} ( \mathbb{R}^{n} ) \) without much change, but we chose the framework of \(\mathcal{S} ( \mathbb{R}^{n} ) \) and \(\mathcal{S}^{\prime} ( \mathbb{R}^{n} ) \) because we will study Fourier transforms.

  3. We shall usually consider the strong topology on dual spaces, but most results also hold for the weak topology as well since in spaces of distributions weak and strong convergence of sequences are equivalent [10, 12].

  4. A proof can be given following the ideas of the proof of Proposition 3.4.

  5. For properties of LF spaces see [10] or [12].

  6. See [2, 7] for the theory of order relation in the Cesàro sense for distributions.

References

  1. Bowen, J.M.: Delta function terms arising from classical point-source fields. Am. J. Phys. 62, 511–515 (1994)

    Article  Google Scholar 

  2. Campos Ferreira, J.: Introduction to the Theory of Distributions. Longman, London (1997)

    MATH  Google Scholar 

  3. Erdélyi, A., Köber, H.: Some remarks on Hankel transforms. Q. J. Math. Oxford Ser. 11, 212–221 (1940)

    Article  Google Scholar 

  4. Estrada, R.: The non-existence of regularization operators. J. Math. Anal. Appl. 286, 1–10 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Estrada, R., Kanwal, R.P.: Regularization and distributional derivatives of \(( x_{1}^{2}+\cdots+ x_{p}^{2}) ^{n/2}\) in \(\mathcal{D}^{\prime}( \mathbb{R}^{p})\). Proc. R. Soc. Lond. A 401, 281–297 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  6. Estrada, R., Kanwal, R.P.: Singular Integral Equations. Birkhäuser, Boston (2000)

    Book  MATH  Google Scholar 

  7. Estrada, R., Kanwal, R.P.: A Distributional Approach to Asymptotics. Theory and Applications, 2nd edn. Birkhäuser, Boston (2002)

    Book  MATH  Google Scholar 

  8. Farassat, F.: Introduction to generalized functions with applications in aerodynamics and aeroacoustics. NASA Technical Paper 3248, Hampton, VA: NASA Langley Research Center (1996). http://ntrs.nasa.gov

  9. Grafakos, L., Teschl, G.: On Fourier transforms of radial functions and distributions. J. Fourier Anal. Appl. 19, 167–179 (2013)

    Article  MathSciNet  Google Scholar 

  10. Horváth, J.: Topological Vector Spaces and Distributions, vol. I. Addison-Wesley, Reading (1966)

    MATH  Google Scholar 

  11. Kanwal, R.P.: Generalized Functions: Theory and Technique, 3rd edn. Birkhäuser, Boston (2004)

    Book  Google Scholar 

  12. Trèves, F.: Topological Vector Spaces, Distributions, and Kernels. Academic Press, New York (1967)

    MATH  Google Scholar 

  13. Walton, J.R.: A distributional approach to dual integral equations of Titchmarsh type. SIAM J. Math. Anal. 6, 628–643 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  14. Zemanian, A.H.: Generalized Integral Transforms. Wiley, New York (1965)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Ricardo Estrada

Authors
  1. Ricardo Estrada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo Estrada.

Additional information

Communicated by Loukas Grafakos.

The author gratefully acknowledges support from NSF, through grant number 0968448.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Estrada, R. On Radial Functions and Distributions and Their Fourier Transforms. J Fourier Anal Appl 20, 301–320 (2014). https://doi.org/10.1007/s00041-013-9313-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-013-9313-2

Keywords

Mathematics Subject Classification

Search

Navigation